LMFDB - Embedded newform 264.3.p.b.131.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [264,3,Mod(131,264)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(264, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("264.131"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 264 = 2^{3} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	264.p (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.19347897911$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			131.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	264.131
Dual form			264.3.p.b.131.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +(1.00000 + 2.82843i) q^{3} +4.00000 q^{4} +(-2.00000 - 5.65685i) q^{6} -8.00000 q^{8} +(-7.00000 + 5.65685i) q^{9} +(-7.00000 + 8.48528i) q^{11} +(4.00000 + 11.3137i) q^{12} +16.0000 q^{16} -2.00000 q^{17} +(14.0000 - 11.3137i) q^{18} +16.9706i q^{19} +(14.0000 - 16.9706i) q^{22} +(-8.00000 - 22.6274i) q^{24} -25.0000 q^{25} +(-23.0000 - 14.1421i) q^{27} -32.0000 q^{32} +(-31.0000 - 11.3137i) q^{33} +4.00000 q^{34} +(-28.0000 + 22.6274i) q^{36} -33.9411i q^{38} -46.0000 q^{41} +84.8528i q^{43} +(-28.0000 + 33.9411i) q^{44} +(16.0000 + 45.2548i) q^{48} -49.0000 q^{49} +50.0000 q^{50} +(-2.00000 - 5.65685i) q^{51} +(46.0000 + 28.2843i) q^{54} +(-48.0000 + 16.9706i) q^{57} -84.8528i q^{59} +64.0000 q^{64} +(62.0000 + 22.6274i) q^{66} +62.0000 q^{67} -8.00000 q^{68} +(56.0000 - 45.2548i) q^{72} +33.9411i q^{73} +(-25.0000 - 70.7107i) q^{75} +67.8823i q^{76} +(17.0000 - 79.1960i) q^{81} +92.0000 q^{82} +158.000 q^{83} -169.706i q^{86} +(56.0000 - 67.8823i) q^{88} +101.823i q^{89} +(-32.0000 - 90.5097i) q^{96} +94.0000 q^{97} +98.0000 q^{98} +(1.00000 - 98.9949i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{6} - 16 q^{8} - 14 q^{9} - 14 q^{11} + 8 q^{12} + 32 q^{16} - 4 q^{17} + 28 q^{18} + 28 q^{22} - 16 q^{24} - 50 q^{25} - 46 q^{27} - 64 q^{32} - 62 q^{33} + 8 q^{34}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 4 * q^6 - 16 * q^8 - 14 * q^9 - 14 * q^11 + 8 * q^12 + 32 * q^16 - 4 * q^17 + 28 * q^18 + 28 * q^22 - 16 * q^24 - 50 * q^25 - 46 * q^27 - 64 * q^32 - 62 * q^33 + 8 * q^34 - 56 * q^36 - 92 * q^41 - 56 * q^44 + 32 * q^48 - 98 * q^49 + 100 * q^50 - 4 * q^51 + 92 * q^54 - 96 * q^57 + 128 * q^64 + 124 * q^66 + 124 * q^67 - 16 * q^68 + 112 * q^72 - 50 * q^75 + 34 * q^81 + 184 * q^82 + 316 * q^83 + 112 * q^88 - 64 * q^96 + 188 * q^97 + 196 * q^98 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/264\mathbb{Z}\right)^\times$.

$n$	$89$	$133$	$145$	$199$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.00000			−1.00000
$2$	−2.00000			−1.00000
$3$	1.00000	+	2.82843i	0.333333	+	0.942809i
$3$	1.00000	+	2.82843i	0.333333	+	0.942809i
$4$	4.00000			1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−2.00000	−	5.65685i	−0.333333	−	0.942809i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−8.00000			−1.00000
$9$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$10$		0			0
$11$	−7.00000	+	8.48528i	−0.636364	+	0.771389i
$11$	−7.00000	+	8.48528i	−0.636364	+	0.771389i
$12$	4.00000	+	11.3137i	0.333333	+	0.942809i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	16.0000			1.00000
$17$	−2.00000			−0.117647			−0.0588235	−	0.998268i	$-0.518735\pi$
$17$	−2.00000			−0.117647			−0.0588235	+	0.998268i	$0.518735\pi$
$18$	14.0000	−	11.3137i	0.777778	−	0.628539i
$19$			16.9706i			0.893188i	0.894737	+	0.446594i	$0.147363\pi$
$19$			16.9706i			0.893188i	−0.894737	+	0.446594i	$0.852637\pi$
$20$		0			0
$21$		0			0
$22$	14.0000	−	16.9706i	0.636364	−	0.771389i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−8.00000	−	22.6274i	−0.333333	−	0.942809i
$25$	−25.0000			−1.00000
$26$		0			0
$27$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−32.0000			−1.00000
$33$	−31.0000	−	11.3137i	−0.939394	−	0.342840i
$34$	4.00000			0.117647
$35$		0			0
$36$	−28.0000	+	22.6274i	−0.777778	+	0.628539i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	33.9411i		−	0.893188i
$39$		0			0
$40$		0			0
$41$	−46.0000			−1.12195			−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−46.0000			−1.12195			−0.560976	+	0.827832i	$0.689574\pi$
$42$		0			0
$43$			84.8528i			1.97332i	0.162791	+	0.986661i	$0.447950\pi$
$43$			84.8528i			1.97332i	−0.162791	+	0.986661i	$0.552050\pi$
$44$	−28.0000	+	33.9411i	−0.636364	+	0.771389i
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	16.0000	+	45.2548i	0.333333	+	0.942809i
$49$	−49.0000			−1.00000
$50$	50.0000			1.00000
$51$	−2.00000	−	5.65685i	−0.0392157	−	0.110919i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	46.0000	+	28.2843i	0.851852	+	0.523783i
$55$		0			0
$56$		0			0
$57$	−48.0000	+	16.9706i	−0.842105	+	0.297729i
$58$		0			0
$59$		−	84.8528i		−	1.43818i	−0.694915	−	0.719092i	$-0.744558\pi$
$59$		−	84.8528i		−	1.43818i	0.694915	−	0.719092i	$-0.255442\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$		0			0
$66$	62.0000	+	22.6274i	0.939394	+	0.342840i
$67$	62.0000			0.925373			0.462687	−	0.886522i	$-0.346886\pi$
$67$	62.0000			0.925373			0.462687	+	0.886522i	$0.346886\pi$
$68$	−8.00000			−0.117647
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	56.0000	−	45.2548i	0.777778	−	0.628539i
$73$			33.9411i			0.464947i	0.972603	+	0.232473i	$0.0746819\pi$
$73$			33.9411i			0.464947i	−0.972603	+	0.232473i	$0.925318\pi$
$74$		0			0
$75$	−25.0000	−	70.7107i	−0.333333	−	0.942809i
$76$			67.8823i			0.893188i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	17.0000	−	79.1960i	0.209877	−	0.977728i
$82$	92.0000			1.12195
$83$	158.000			1.90361			0.951807	−	0.306697i	$-0.0992238\pi$
$83$	158.000			1.90361			0.951807	+	0.306697i	$0.0992238\pi$
$84$		0			0
$85$		0			0
$86$		−	169.706i		−	1.97332i
$87$		0			0
$88$	56.0000	−	67.8823i	0.636364	−	0.771389i
$89$			101.823i			1.14408i	0.820225	+	0.572041i	$0.193848\pi$
$89$			101.823i			1.14408i	−0.820225	+	0.572041i	$0.806152\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−32.0000	−	90.5097i	−0.333333	−	0.942809i
$97$	94.0000			0.969072			0.484536	−	0.874771i	$-0.338988\pi$
$97$	94.0000			0.969072			0.484536	+	0.874771i	$0.338988\pi$
$98$	98.0000			1.00000
$99$	1.00000	−	98.9949i	0.0101010	−	0.999949i
$100$	−100.000			−1.00000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	4.00000	+	11.3137i	0.0392157	+	0.110919i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	178.000			1.66355			0.831776	−	0.555112i	$-0.187325\pi$
$107$	178.000			1.66355			0.831776	+	0.555112i	$0.187325\pi$
$108$	−92.0000	−	56.5685i	−0.851852	−	0.523783i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			203.647i			1.80218i	0.433628	+	0.901092i	$0.357233\pi$
$113$			203.647i			1.80218i	−0.433628	+	0.901092i	$0.642767\pi$
$114$	96.0000	−	33.9411i	0.842105	−	0.297729i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			169.706i			1.43818i
$119$		0			0
$120$		0			0
$121$	−23.0000	−	118.794i	−0.190083	−	0.981768i
$122$		0			0
$123$	−46.0000	−	130.108i	−0.373984	−	1.05779i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−128.000			−1.00000
$129$	−240.000	+	84.8528i	−1.86047	+	0.657774i
$130$		0			0
$131$	62.0000			0.473282			0.236641	−	0.971597i	$-0.423953\pi$
$131$	62.0000			0.473282			0.236641	+	0.971597i	$0.423953\pi$
$132$	−124.000	−	45.2548i	−0.939394	−	0.342840i
$133$		0			0
$134$	−124.000			−0.925373
$135$		0			0
$136$	16.0000			0.117647
$137$			135.765i			0.990982i	0.868613	+	0.495491i	$0.165012\pi$
$137$			135.765i			0.990982i	−0.868613	+	0.495491i	$0.834988\pi$
$138$		0			0
$139$			186.676i			1.34299i	0.741007	+	0.671497i	$0.234348\pi$
$139$			186.676i			1.34299i	−0.741007	+	0.671497i	$0.765652\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−112.000	+	90.5097i	−0.777778	+	0.628539i
$145$		0			0
$146$		−	67.8823i		−	0.464947i
$147$	−49.0000	−	138.593i	−0.333333	−	0.942809i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	50.0000	+	141.421i	0.333333	+	0.942809i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		−	135.765i		−	0.893188i
$153$	14.0000	−	11.3137i	0.0915033	−	0.0739458i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−34.0000	+	158.392i	−0.209877	+	0.977728i
$163$	322.000			1.97546			0.987730	−	0.156171i	$-0.0499150\pi$
$163$	322.000			1.97546			0.987730	+	0.156171i	$0.0499150\pi$
$164$	−184.000			−1.12195
$165$		0			0
$166$	−316.000			−1.90361
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$	−96.0000	−	118.794i	−0.561404	−	0.694701i
$172$			339.411i			1.97332i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	−112.000	+	135.765i	−0.636364	+	0.771389i
$177$	240.000	−	84.8528i	1.35593	−	0.479394i
$178$		−	203.647i		−	1.14408i
$179$			356.382i			1.99096i	0.0949721	+	0.995480i	$0.469724\pi$
$179$			356.382i			1.99096i	−0.0949721	+	0.995480i	$0.530276\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	14.0000	−	16.9706i	0.0748663	−	0.0907517i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	64.0000	+	181.019i	0.333333	+	0.942809i
$193$		−	373.352i		−	1.93447i	−0.253886	−	0.967234i	$-0.581709\pi$
$193$		−	373.352i		−	1.93447i	0.253886	−	0.967234i	$-0.418291\pi$
$194$	−188.000			−0.969072
$195$		0			0
$196$	−196.000			−1.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−2.00000	+	197.990i	−0.0101010	+	0.999949i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	200.000			1.00000
$201$	62.0000	+	175.362i	0.308458	+	0.872450i
$202$		0			0
$203$		0			0
$204$	−8.00000	−	22.6274i	−0.0392157	−	0.110919i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−144.000	−	118.794i	−0.688995	−	0.568392i
$210$		0			0
$211$		−	356.382i		−	1.68901i	−0.535545	−	0.844507i	$-0.679894\pi$
$211$		−	356.382i		−	1.68901i	0.535545	−	0.844507i	$-0.320106\pi$
$212$		0			0
$213$		0			0
$214$	−356.000			−1.66355
$215$		0			0
$216$	184.000	+	113.137i	0.851852	+	0.523783i
$217$		0			0
$218$		0			0
$219$	−96.0000	+	33.9411i	−0.438356	+	0.154982i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	175.000	−	141.421i	0.777778	−	0.628539i
$226$		−	407.294i		−	1.80218i
$227$	−446.000			−1.96476			−0.982379	−	0.186900i	$-0.940156\pi$
$227$	−446.000			−1.96476			−0.982379	+	0.186900i	$0.940156\pi$
$228$	−192.000	+	67.8823i	−0.842105	+	0.297729i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	434.000			1.86266			0.931330	−	0.364175i	$-0.118649\pi$
$233$	434.000			1.86266			0.931330	+	0.364175i	$0.118649\pi$
$234$		0			0
$235$		0			0
$236$		−	339.411i		−	1.43818i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	441.235i		−	1.83085i	−0.402490	−	0.915425i	$-0.631855\pi$
$241$		−	441.235i		−	1.83085i	0.402490	−	0.915425i	$-0.368145\pi$
$242$	46.0000	+	237.588i	0.190083	+	0.981768i
$243$	241.000	−	31.1127i	0.991770	−	0.128036i
$244$		0			0
$245$		0			0
$246$	92.0000	+	260.215i	0.373984	+	1.05779i
$247$		0			0
$248$		0			0
$249$	158.000	+	446.891i	0.634538	+	1.79474i
$250$		0			0
$251$			186.676i			0.743730i	0.928287	+	0.371865i	$0.121282\pi$
$251$			186.676i			0.743730i	−0.928287	+	0.371865i	$0.878718\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	256.000			1.00000
$257$		−	339.411i		−	1.32067i	−0.750973	−	0.660333i	$-0.770415\pi$
$257$		−	339.411i		−	1.32067i	0.750973	−	0.660333i	$-0.229585\pi$
$258$	480.000	−	169.706i	1.86047	−	0.657774i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−124.000			−0.473282
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	248.000	+	90.5097i	0.939394	+	0.342840i
$265$		0			0
$266$		0			0
$267$	−288.000	+	101.823i	−1.07865	+	0.381361i
$268$	248.000			0.925373
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−32.0000			−0.117647
$273$		0			0
$274$		−	271.529i		−	0.990982i
$275$	175.000	−	212.132i	0.636364	−	0.771389i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		−	373.352i		−	1.34299i
$279$		0			0
$280$		0			0
$281$	238.000			0.846975			0.423488	−	0.905902i	$-0.360806\pi$
$281$	238.000			0.846975			0.423488	+	0.905902i	$0.360806\pi$
$282$		0			0
$283$			560.029i			1.97890i	0.144876	+	0.989450i	$0.453722\pi$
$283$			560.029i			1.97890i	−0.144876	+	0.989450i	$0.546278\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	224.000	−	181.019i	0.777778	−	0.628539i
$289$	−285.000			−0.986159
$290$		0			0
$291$	94.0000	+	265.872i	0.323024	+	0.913650i
$292$			135.765i			0.464947i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	98.0000	+	277.186i	0.333333	+	0.942809i
$295$		0			0
$296$		0			0
$297$	281.000	−	96.1665i	0.946128	−	0.323793i
$298$		0			0
$299$		0			0
$300$	−100.000	−	282.843i	−0.333333	−	0.942809i
$301$		0			0
$302$		0			0
$303$		0			0
$304$			271.529i			0.893188i
$305$		0			0
$306$	−28.0000	+	22.6274i	−0.0915033	+	0.0739458i
$307$			288.500i			0.939738i	0.882736	+	0.469869i	$0.155699\pi$
$307$			288.500i			0.939738i	−0.882736	+	0.469869i	$0.844301\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−526.000			−1.68051			−0.840256	−	0.542191i	$-0.817595\pi$
$313$	−526.000			−1.68051			−0.840256	+	0.542191i	$0.817595\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	178.000	+	503.460i	0.554517	+	1.56841i
$322$		0			0
$323$		−	33.9411i		−	0.105081i
$324$	68.0000	−	316.784i	0.209877	−	0.977728i
$325$		0			0
$326$	−644.000			−1.97546
$327$		0			0
$328$	368.000			1.12195
$329$		0			0
$330$		0			0
$331$	−14.0000			−0.0422961			−0.0211480	−	0.999776i	$-0.506732\pi$
$331$	−14.0000			−0.0422961			−0.0211480	+	0.999776i	$0.506732\pi$
$332$	632.000			1.90361
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			475.176i			1.41002i	0.709199	+	0.705009i	$0.249057\pi$
$337$			475.176i			1.41002i	−0.709199	+	0.705009i	$0.750943\pi$
$338$	338.000			1.00000
$339$	−576.000	+	203.647i	−1.69912	+	0.600728i
$340$		0			0
$341$		0			0
$342$	192.000	+	237.588i	0.561404	+	0.694701i
$343$		0			0
$344$		−	678.823i		−	1.97332i
$345$		0			0
$346$		0			0
$347$	−658.000			−1.89625			−0.948127	−	0.317892i	$-0.897025\pi$
$347$	−658.000			−1.89625			−0.948127	+	0.317892i	$0.897025\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	224.000	−	271.529i	0.636364	−	0.771389i
$353$		−	678.823i		−	1.92301i	−0.274788	−	0.961505i	$-0.588608\pi$
$353$		−	678.823i		−	1.92301i	0.274788	−	0.961505i	$-0.411392\pi$
$354$	−480.000	+	169.706i	−1.35593	+	0.479394i
$355$		0			0
$356$			407.294i			1.14408i
$357$		0			0
$358$		−	712.764i		−	1.99096i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	73.0000			0.202216
$362$		0			0
$363$	313.000	−	183.848i	0.862259	−	0.506468i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	322.000	−	260.215i	0.872629	−	0.705191i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−28.0000	+	33.9411i	−0.0748663	+	0.0907517i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−686.000			−1.81003			−0.905013	−	0.425383i	$-0.860139\pi$
$379$	−686.000			−1.81003			−0.905013	+	0.425383i	$0.860139\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−128.000	−	362.039i	−0.333333	−	0.942809i
$385$		0			0
$386$			746.705i			1.93447i
$387$	−480.000	−	593.970i	−1.24031	−	1.53481i
$388$	376.000			0.969072
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	392.000			1.00000
$393$	62.0000	+	175.362i	0.157761	+	0.446215i
$394$		0			0
$395$		0			0
$396$	4.00000	−	395.980i	0.0101010	−	0.999949i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−400.000			−1.00000
$401$			237.588i			0.592488i	0.955112	+	0.296244i	$0.0957342\pi$
$401$			237.588i			0.592488i	−0.955112	+	0.296244i	$0.904266\pi$
$402$	−124.000	−	350.725i	−0.308458	−	0.872450i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	16.0000	+	45.2548i	0.0392157	+	0.110919i
$409$			746.705i			1.82568i	0.408313	+	0.912842i	$0.366117\pi$
$409$			746.705i			1.82568i	−0.408313	+	0.912842i	$0.633883\pi$
$410$		0			0
$411$	−384.000	+	135.765i	−0.934307	+	0.330327i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−528.000	+	186.676i	−1.26619	+	0.447665i
$418$	288.000	+	237.588i	0.688995	+	0.568392i
$419$		−	661.852i		−	1.57960i	−0.613365	−	0.789799i	$-0.710185\pi$
$419$		−	661.852i		−	1.57960i	0.613365	−	0.789799i	$-0.289815\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			712.764i			1.68901i
$423$		0			0
$424$		0			0
$425$	50.0000			0.117647
$426$		0			0
$427$		0			0
$428$	712.000			1.66355
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−368.000	−	226.274i	−0.851852	−	0.523783i
$433$	578.000			1.33487			0.667436	−	0.744667i	$-0.267392\pi$
$433$	578.000			1.33487			0.667436	+	0.744667i	$0.267392\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	192.000	−	67.8823i	0.438356	−	0.154982i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	343.000	−	277.186i	0.777778	−	0.628539i
$442$		0			0
$443$			118.794i			0.268158i	0.990971	+	0.134079i	$0.0428076\pi$
$443$			118.794i			0.268158i	−0.990971	+	0.134079i	$0.957192\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			237.588i			0.529149i	0.964365	+	0.264574i	$0.0852315\pi$
$449$			237.588i			0.529149i	−0.964365	+	0.264574i	$0.914769\pi$
$450$	−350.000	+	282.843i	−0.777778	+	0.628539i
$451$	322.000	−	390.323i	0.713969	−	0.865461i
$452$			814.587i			1.80218i
$453$		0			0
$454$	892.000			1.96476
$455$		0			0
$456$	384.000	−	135.765i	0.842105	−	0.297729i
$457$		−	882.469i		−	1.93100i	−0.260394	−	0.965502i	$-0.583852\pi$
$457$		−	882.469i		−	1.93100i	0.260394	−	0.965502i	$-0.416148\pi$
$458$		0			0
$459$	46.0000	+	28.2843i	0.100218	+	0.0616215i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−868.000			−1.86266
$467$			933.381i			1.99867i	0.0364026	+	0.999337i	$0.488410\pi$
$467$			933.381i			1.99867i	−0.0364026	+	0.999337i	$0.511590\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$			678.823i			1.43818i
$473$	−720.000	−	593.970i	−1.52220	−	1.25575i
$474$		0			0
$475$		−	424.264i		−	0.893188i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$			882.469i			1.83085i
$483$		0			0
$484$	−92.0000	−	475.176i	−0.190083	−	0.981768i
$485$		0			0
$486$	−482.000	+	62.2254i	−0.991770	+	0.128036i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	322.000	+	910.754i	0.658487	+	1.86248i
$490$		0			0
$491$	−782.000			−1.59267			−0.796334	−	0.604857i	$-0.793230\pi$
$491$	−782.000			−1.59267			−0.796334	+	0.604857i	$0.793230\pi$
$492$	−184.000	−	520.431i	−0.373984	−	1.05779i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−316.000	−	893.783i	−0.634538	−	1.79474i
$499$	802.000			1.60721			0.803607	−	0.595160i	$-0.202911\pi$
$499$	802.000			1.60721			0.803607	+	0.595160i	$0.202911\pi$
$500$		0			0
$501$		0			0
$502$		−	373.352i		−	0.743730i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−169.000	−	478.004i	−0.333333	−	0.942809i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−1.00000
$513$	240.000	−	390.323i	0.467836	−	0.760863i
$514$			678.823i			1.32067i
$515$		0			0
$516$	−960.000	+	339.411i	−1.86047	+	0.657774i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			271.529i			0.521169i	0.965451	+	0.260584i	$0.0839152\pi$
$521$			271.529i			0.521169i	−0.965451	+	0.260584i	$0.916085\pi$
$522$		0			0
$523$			967.322i			1.84956i	0.380497	+	0.924782i	$0.375753\pi$
$523$			967.322i			1.84956i	−0.380497	+	0.924782i	$0.624247\pi$
$524$	248.000			0.473282
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−496.000	−	181.019i	−0.939394	−	0.342840i
$529$	−529.000			−1.00000
$530$		0			0
$531$	480.000	+	593.970i	0.903955	+	1.11859i
$532$		0			0
$533$		0			0
$534$	576.000	−	203.647i	1.07865	−	0.381361i
$535$		0			0
$536$	−496.000			−0.925373
$537$	−1008.00	+	356.382i	−1.87709	+	0.663653i
$538$		0			0
$539$	343.000	−	415.779i	0.636364	−	0.771389i
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$	64.0000			0.117647
$545$		0			0
$546$		0			0
$547$		−	390.323i		−	0.713570i	−0.934186	−	0.356785i	$-0.883873\pi$
$547$		−	390.323i		−	0.713570i	0.934186	−	0.356785i	$-0.116127\pi$
$548$			543.058i			0.990982i
$549$		0			0
$550$	−350.000	+	424.264i	−0.636364	+	0.771389i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$			746.705i			1.34299i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	62.0000	+	22.6274i	0.110517	+	0.0403341i
$562$	−476.000			−0.846975
$563$	226.000			0.401421			0.200710	−	0.979651i	$-0.435675\pi$
$563$	226.000			0.401421			0.200710	+	0.979651i	$0.435675\pi$
$564$		0			0
$565$		0			0
$566$		−	1120.06i		−	1.97890i
$567$		0			0
$568$		0			0
$569$	−626.000			−1.10018			−0.550088	−	0.835107i	$-0.685406\pi$
$569$	−626.000			−1.10018			−0.550088	+	0.835107i	$0.685406\pi$
$570$		0			0
$571$			933.381i			1.63464i	0.576182	+	0.817321i	$0.304542\pi$
$571$			933.381i			1.63464i	−0.576182	+	0.817321i	$0.695458\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−448.000	+	362.039i	−0.777778	+	0.628539i
$577$	−2.00000			−0.00346620			−0.00173310	−	0.999998i	$-0.500552\pi$
$577$	−2.00000			−0.00346620			−0.00173310	+	0.999998i	$0.500552\pi$
$578$	570.000			0.986159
$579$	1056.00	−	373.352i	1.82383	−	0.644823i
$580$		0			0
$581$		0			0
$582$	−188.000	−	531.744i	−0.323024	−	0.913650i
$583$		0			0
$584$		−	271.529i		−	0.464947i
$585$		0			0
$586$		0			0
$587$		−	288.500i		−	0.491481i	−0.969336	−	0.245741i	$-0.920969\pi$
$587$		−	288.500i		−	0.491481i	0.969336	−	0.245741i	$-0.0790312\pi$
$588$	−196.000	−	554.372i	−0.333333	−	0.942809i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	862.000			1.45363			0.726813	−	0.686836i	$-0.241001\pi$
$593$	862.000			1.45363			0.726813	+	0.686836i	$0.241001\pi$
$594$	−562.000	+	192.333i	−0.946128	+	0.323793i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	200.000	+	565.685i	0.333333	+	0.942809i
$601$		−	780.646i		−	1.29891i	−0.760399	−	0.649456i	$-0.774997\pi$
$601$		−	780.646i		−	1.29891i	0.760399	−	0.649456i	$-0.225003\pi$
$602$		0			0
$603$	−434.000	+	350.725i	−0.719735	+	0.581633i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		−	543.058i		−	0.893188i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	56.0000	−	45.2548i	0.0915033	−	0.0739458i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		−	576.999i		−	0.939738i
$615$		0			0
$616$		0			0
$617$		−	1187.94i		−	1.92535i	−0.270665	−	0.962674i	$-0.587243\pi$
$617$		−	1187.94i		−	1.92535i	0.270665	−	0.962674i	$-0.412757\pi$
$618$		0			0
$619$	−562.000			−0.907916			−0.453958	−	0.891023i	$-0.649988\pi$
$619$	−562.000			−0.907916			−0.453958	+	0.891023i	$0.649988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	625.000			1.00000
$626$	1052.00			1.68051
$627$	192.000	−	526.087i	0.306220	−	0.839055i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	1008.00	−	356.382i	1.59242	−	0.563004i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1187.94i			1.85326i	0.375975	+	0.926630i	$0.377308\pi$
$641$			1187.94i			1.85326i	−0.375975	+	0.926630i	$0.622692\pi$
$642$	−356.000	−	1006.92i	−0.554517	−	1.56841i
$643$	1214.00			1.88802			0.944012	−	0.329910i	$-0.107018\pi$
$643$	1214.00			1.88802			0.944012	+	0.329910i	$0.107018\pi$
$644$		0			0
$645$		0			0
$646$			67.8823i			0.105081i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−136.000	+	633.568i	−0.209877	+	0.977728i
$649$	720.000	+	593.970i	1.10940	+	0.915208i
$650$		0			0
$651$		0			0
$652$	1288.00			1.97546
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$	−736.000			−1.12195
$657$	−192.000	−	237.588i	−0.292237	−	0.361625i
$658$		0			0
$659$	994.000			1.50835			0.754173	−	0.656676i	$-0.228038\pi$
$659$	994.000			1.50835			0.754173	+	0.656676i	$0.228038\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	28.0000			0.0422961
$663$		0			0
$664$	−1264.00			−1.90361
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			509.117i			0.756489i	0.925706	+	0.378244i	$0.123472\pi$
$673$			509.117i			0.756489i	−0.925706	+	0.378244i	$0.876528\pi$
$674$		−	950.352i		−	1.41002i
$675$	575.000	+	353.553i	0.851852	+	0.523783i
$676$	−676.000			−1.00000
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	1152.00	−	407.294i	1.69912	−	0.600728i
$679$		0			0
$680$		0			0
$681$	−446.000	−	1261.48i	−0.654919	−	1.85239i
$682$		0			0
$683$		−	1306.73i		−	1.91323i	−0.291362	−	0.956613i	$-0.594108\pi$
$683$		−	1306.73i		−	1.91323i	0.291362	−	0.956613i	$-0.405892\pi$
$684$	−384.000	−	475.176i	−0.561404	−	0.694701i
$685$		0			0
$686$		0			0
$687$		0			0
$688$			1357.65i			1.97332i
$689$		0			0
$690$		0			0
$691$	734.000			1.06223			0.531114	−	0.847300i	$-0.321773\pi$
$691$	734.000			1.06223			0.531114	+	0.847300i	$0.321773\pi$
$692$		0			0
$693$		0			0
$694$	1316.00			1.89625
$695$		0			0
$696$		0			0
$697$	92.0000			0.131994
$698$		0			0
$699$	434.000	+	1227.54i	0.620887	+	1.75613i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−448.000	+	543.058i	−0.636364	+	0.771389i
$705$		0			0
$706$			1357.65i			1.92301i
$707$		0			0
$708$	960.000	−	339.411i	1.35593	−	0.479394i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		−	814.587i		−	1.14408i
$713$		0			0
$714$		0			0
$715$		0			0
$716$			1425.53i			1.99096i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−146.000			−0.202216
$723$	1248.00	−	441.235i	1.72614	−	0.610283i
$724$		0			0
$725$		0			0
$726$	−626.000	+	367.696i	−0.862259	+	0.506468i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$		−	169.706i		−	0.232155i
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−434.000	+	526.087i	−0.588874	+	0.713823i
$738$	−644.000	+	520.431i	−0.872629	+	0.705191i
$739$		−	1442.50i		−	1.95196i	−0.217862	−	0.975980i	$-0.569908\pi$
$739$		−	1442.50i		−	1.95196i	0.217862	−	0.975980i	$-0.430092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1106.00	+	893.783i	−1.48059	+	1.19650i
$748$	56.0000	−	67.8823i	0.0748663	−	0.0907517i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−528.000	+	186.676i	−0.701195	+	0.247910i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1372.00			1.81003
$759$		0			0
$760$		0			0
$761$	−1394.00			−1.83180			−0.915900	−	0.401406i	$-0.868522\pi$
$761$	−1394.00			−1.83180			−0.915900	+	0.401406i	$0.868522\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	256.000	+	724.077i	0.333333	+	0.942809i
$769$		−	1120.06i		−	1.45651i	−0.685306	−	0.728256i	$-0.740331\pi$
$769$		−	1120.06i		−	1.45651i	0.685306	−	0.728256i	$-0.259669\pi$
$770$		0			0
$771$	960.000	−	339.411i	1.24514	−	0.440222i
$772$		−	1493.41i		−	1.93447i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	960.000	+	1187.94i	1.24031	+	1.53481i
$775$		0			0
$776$	−752.000			−0.969072
$777$		0			0
$778$		0			0
$779$		−	780.646i		−	1.00211i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−784.000			−1.00000
$785$		0			0
$786$	−124.000	−	350.725i	−0.157761	−	0.446215i
$787$			1272.79i			1.61727i	0.588310	+	0.808635i	$0.299793\pi$
$787$			1272.79i			1.61727i	−0.588310	+	0.808635i	$0.700207\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−8.00000	+	791.960i	−0.0101010	+	0.999949i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	800.000			1.00000
$801$	−576.000	−	712.764i	−0.719101	−	0.889842i
$802$		−	475.176i		−	0.592488i
$803$	−288.000	−	237.588i	−0.358655	−	0.295875i
$804$	248.000	+	701.450i	0.308458	+	0.872450i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1582.00			−1.95550			−0.977750	−	0.209772i	$-0.932728\pi$
$809$	−1582.00			−1.95550			−0.977750	+	0.209772i	$0.932728\pi$
$810$		0			0
$811$		−	1612.20i		−	1.98792i	−0.109741	−	0.993960i	$-0.535002\pi$
$811$		−	1612.20i		−	1.98792i	0.109741	−	0.993960i	$-0.464998\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−32.0000	−	90.5097i	−0.0392157	−	0.110919i
$817$	−1440.00			−1.76255
$818$		−	1493.41i		−	1.82568i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	768.000	−	271.529i	0.934307	−	0.330327i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	775.000	+	282.843i	0.939394	+	0.342840i
$826$		0			0
$827$	1262.00			1.52600			0.762999	−	0.646400i	$-0.223726\pi$
$827$	1262.00			1.52600			0.762999	+	0.646400i	$0.223726\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	98.0000			0.117647
$834$	1056.00	−	373.352i	1.26619	−	0.447665i
$835$		0			0
$836$	−576.000	−	475.176i	−0.688995	−	0.568392i
$837$		0			0
$838$			1323.70i			1.57960i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$	238.000	+	673.166i	0.282325	+	0.798536i
$844$		−	1425.53i		−	1.68901i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1584.00	+	560.029i	−1.86572	+	0.659633i
$850$	−100.000			−0.117647
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−1424.00			−1.66355
$857$	1202.00			1.40257			0.701284	−	0.712882i	$-0.252611\pi$
$857$	1202.00			1.40257			0.701284	+	0.712882i	$0.252611\pi$
$858$		0			0
$859$	−1646.00			−1.91618			−0.958091	−	0.286465i	$-0.907520\pi$
$859$	−1646.00			−1.91618			−0.958091	+	0.286465i	$0.907520\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	736.000	+	452.548i	0.851852	+	0.523783i
$865$		0			0
$866$	−1156.00			−1.33487
$867$	−285.000	−	806.102i	−0.328720	−	0.929760i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−658.000	+	531.744i	−0.753723	+	0.609100i
$874$		0			0
$875$		0			0
$876$	−384.000	+	135.765i	−0.438356	+	0.154982i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1018.23i			1.15577i	0.816118	+	0.577885i	$0.196122\pi$
$881$			1018.23i			1.15577i	−0.816118	+	0.577885i	$0.803878\pi$
$882$	−686.000	+	554.372i	−0.777778	+	0.628539i
$883$	−1762.00			−1.99547			−0.997735	−	0.0672672i	$-0.978572\pi$
$883$	−1762.00			−1.99547			−0.997735	+	0.0672672i	$0.978572\pi$
$884$		0			0
$885$		0			0
$886$		−	237.588i		−	0.268158i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	553.000	+	698.621i	0.620651	+	0.784087i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		−	475.176i		−	0.529149i
$899$		0			0
$900$	700.000	−	565.685i	0.777778	−	0.628539i
$901$		0			0
$902$	−644.000	+	780.646i	−0.713969	+	0.865461i
$903$		0			0
$904$		−	1629.17i		−	1.80218i
$905$		0			0
$906$		0			0
$907$	−1714.00			−1.88975			−0.944873	−	0.327436i	$-0.893815\pi$
$907$	−1714.00			−1.88975			−0.944873	+	0.327436i	$0.893815\pi$
$908$	−1784.00			−1.96476
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−768.000	+	271.529i	−0.842105	+	0.297729i
$913$	−1106.00	+	1340.67i	−1.21139	+	1.46843i
$914$			1764.94i			1.93100i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−92.0000	−	56.5685i	−0.100218	−	0.0616215i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−816.000	+	288.500i	−0.885993	+	0.313246i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1527.35i			1.64408i	0.569429	+	0.822040i	$0.307164\pi$
$929$			1527.35i			1.64408i	−0.569429	+	0.822040i	$0.692836\pi$
$930$		0			0
$931$		−	831.558i		−	0.893188i
$932$	1736.00			1.86266
$933$		0			0
$934$		−	1866.76i		−	1.99867i
$935$		0			0
$936$		0			0
$937$			1731.00i			1.84738i	0.383138	+	0.923691i	$0.374843\pi$
$937$			1731.00i			1.84738i	−0.383138	+	0.923691i	$0.625157\pi$
$938$		0			0
$939$	−526.000	−	1487.75i	−0.560170	−	1.58440i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		−	1357.65i		−	1.43818i
$945$		0			0
$946$	1440.00	+	1187.94i	1.52220	+	1.25575i
$947$			1612.20i			1.70243i	0.524815	+	0.851216i	$0.324134\pi$
$947$			1612.20i			1.70243i	−0.524815	+	0.851216i	$0.675866\pi$
$948$		0			0
$949$		0			0
$950$			848.528i			0.893188i
$951$		0			0
$952$		0			0
$953$	−142.000			−0.149003			−0.0745016	−	0.997221i	$-0.523737\pi$
$953$	−142.000			−0.149003			−0.0745016	+	0.997221i	$0.523737\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	−1246.00	+	1006.92i	−1.29387	+	1.04561i
$964$		−	1764.94i		−	1.83085i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	184.000	+	950.352i	0.190083	+	0.981768i
$969$	96.0000	−	33.9411i	0.0990712	−	0.0350270i
$970$		0			0
$971$			1680.09i			1.73026i	0.501545	+	0.865132i	$0.332765\pi$
$971$			1680.09i			1.73026i	−0.501545	+	0.865132i	$0.667235\pi$
$972$	964.000	−	124.451i	0.991770	−	0.128036i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	373.352i		−	0.382142i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$		−	373.352i		−	0.382142i	0.981576	−	0.191071i	$-0.0611960\pi$
$978$	−644.000	−	1821.51i	−0.658487	−	1.86248i
$979$	−864.000	−	712.764i	−0.882533	−	0.728053i
$980$		0			0
$981$		0			0
$982$	1564.00			1.59267
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	368.000	+	1040.86i	0.373984	+	1.05779i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−14.0000	−	39.5980i	−0.0140987	−	0.0398771i
$994$		0			0
$995$		0			0
$996$	632.000	+	1787.57i	0.634538	+	1.79474i
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−1604.00			−1.60721
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	264.3.p.b.131.2	yes	2
3.2	odd	2		264.3.p.e.131.1	yes	2
4.3	odd	2		1056.3.p.d.527.1		2
8.3	odd	2	CM	264.3.p.b.131.2	yes	2
8.5	even	2		1056.3.p.d.527.1		2
11.10	odd	2		264.3.p.e.131.2	yes	2
12.11	even	2		1056.3.p.c.527.2		2
24.5	odd	2		1056.3.p.c.527.2		2
24.11	even	2		264.3.p.e.131.1	yes	2
33.32	even	2	inner	264.3.p.b.131.1	✓	2
44.43	even	2		1056.3.p.c.527.1		2
88.21	odd	2		1056.3.p.c.527.1		2
88.43	even	2		264.3.p.e.131.2	yes	2
132.131	odd	2		1056.3.p.d.527.2		2
264.131	odd	2	inner	264.3.p.b.131.1	✓	2
264.197	even	2		1056.3.p.d.527.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.3.p.b.131.1	✓	2	33.32	even	2	inner
264.3.p.b.131.1	✓	2	264.131	odd	2	inner
264.3.p.b.131.2	yes	2	1.1	even	1	trivial
264.3.p.b.131.2	yes	2	8.3	odd	2	CM
264.3.p.e.131.1	yes	2	3.2	odd	2
264.3.p.e.131.1	yes	2	24.11	even	2
264.3.p.e.131.2	yes	2	11.10	odd	2
264.3.p.e.131.2	yes	2	88.43	even	2
1056.3.p.c.527.1		2	44.43	even	2
1056.3.p.c.527.1		2	88.21	odd	2
1056.3.p.c.527.2		2	12.11	even	2
1056.3.p.c.527.2		2	24.5	odd	2
1056.3.p.d.527.1		2	4.3	odd	2
1056.3.p.d.527.1		2	8.5	even	2
1056.3.p.d.527.2		2	132.131	odd	2
1056.3.p.d.527.2		2	264.197	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 264 = 2^{3} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	264.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +(1.00000 + 2.82843i) q^{3} +4.00000 q^{4} +(-2.00000 - 5.65685i) q^{6} -8.00000 q^{8} +(-7.00000 + 5.65685i) q^{9} +(-7.00000 + 8.48528i) q^{11} +(4.00000 + 11.3137i) q^{12} +16.0000 q^{16} -2.00000 q^{17} +(14.0000 - 11.3137i) q^{18} +16.9706i q^{19} +(14.0000 - 16.9706i) q^{22} +(-8.00000 - 22.6274i) q^{24} -25.0000 q^{25} +(-23.0000 - 14.1421i) q^{27} -32.0000 q^{32} +(-31.0000 - 11.3137i) q^{33} +4.00000 q^{34} +(-28.0000 + 22.6274i) q^{36} -33.9411i q^{38} -46.0000 q^{41} +84.8528i q^{43} +(-28.0000 + 33.9411i) q^{44} +(16.0000 + 45.2548i) q^{48} -49.0000 q^{49} +50.0000 q^{50} +(-2.00000 - 5.65685i) q^{51} +(46.0000 + 28.2843i) q^{54} +(-48.0000 + 16.9706i) q^{57} -84.8528i q^{59} +64.0000 q^{64} +(62.0000 + 22.6274i) q^{66} +62.0000 q^{67} -8.00000 q^{68} +(56.0000 - 45.2548i) q^{72} +33.9411i q^{73} +(-25.0000 - 70.7107i) q^{75} +67.8823i q^{76} +(17.0000 - 79.1960i) q^{81} +92.0000 q^{82} +158.000 q^{83} -169.706i q^{86} +(56.0000 - 67.8823i) q^{88} +101.823i q^{89} +(-32.0000 - 90.5097i) q^{96} +94.0000 q^{97} +98.0000 q^{98} +(1.00000 - 98.9949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{6} - 16 q^{8} - 14 q^{9} - 14 q^{11} + 8 q^{12} + 32 q^{16} - 4 q^{17} + 28 q^{18} + 28 q^{22} - 16 q^{24} - 50 q^{25} - 46 q^{27} - 64 q^{32} - 62 q^{33} + 8 q^{34}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 4 * q^6 - 16 * q^8 - 14 * q^9 - 14 * q^11 + 8 * q^12 + 32 * q^16 - 4 * q^17 + 28 * q^18 + 28 * q^22 - 16 * q^24 - 50 * q^25 - 46 * q^27 - 64 * q^32 - 62 * q^33 + 8 * q^34 - 56 * q^36 - 92 * q^41 - 56 * q^44 + 32 * q^48 - 98 * q^49 + 100 * q^50 - 4 * q^51 + 92 * q^54 - 96 * q^57 + 128 * q^64 + 124 * q^66 + 124 * q^67 - 16 * q^68 + 112 * q^72 - 50 * q^75 + 34 * q^81 + 184 * q^82 + 316 * q^83 + 112 * q^88 - 64 * q^96 + 188 * q^97 + 196 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more